In their book "Theory of sets" Bourbaki suggested a general theory of isomorphism.

(See also http://www.tau.ac.il/~corry/publications/articles/pdf/bourbaki-structures.pdf )

The example of an untransportable relation (i.e. formula) in the book involves 2 principal base sets.

Are there examples of untrasportable formulas when we allow only one principal base set?

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    $\begingroup$ Very few people here are familiar with Bourbaki's set theory. You should probably state the definitions here as well. $\endgroup$ – Harry Gindi Jun 19 '10 at 12:14
  • $\begingroup$ A reference is added in the question. $\endgroup$ – Victor Makarov Jun 22 '10 at 13:44

An example of untrasportable sentence, when there is only one principal base set X, may be the following one:

All elements of the set X are finite sets,

Because, by definition, the truth value of a transportable sentence must be preserved under all bijections from the set X. Obviously, there exists a bijection from X to a set Y, where not all elements of Y are finite sets.

A simpler example is "the set X contains the empty set".

There is a paper "Sentences of type theory: the only sentences preserved under isomorphisms" by Victoria Marshall and Rolando Chuaqui - see The Journal of symbolic Logic, vol 56, #3, Sep 1991.

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